with \(\alpha = \max \left|f^{\prime }(u)\right|\)  is used.
Numerical fluxes about \(u\)  are taken by following the alternating principle [1]}
\(\hat{u}_{i+\frac{1}{2}}^h = u^h(x_{i+\frac{1}{2}}^-), \qquad (\widetilde{u}^h_x)_{i+\frac{1}{2}} = u^h_x(x_{i+\frac{1}{2}}^+), \qquad (\check{u}^h_{xx})_{i+\frac{1}{2}} = u^h_{xx}(x_{i+\frac{1}{2}}^+)\) 
